TUNNEL EFFECT
TUNNEL EFFECT
(tunneling), overcoming a potential barrier by a microparticle in the case when its complete (remaining at T.e. for the most part unchanged) less height barrier. That is, the phenomenon is essentially quantum. nature, impossible in classical. mechanics; analogue of T. e. in waves optics can be served by the penetration of light into the reflecting medium (at distances of the order of the light wavelength) in conditions where, from the point of view of geom. optics is happening. T. e. underlies plural important processes in at. and they say physics, in physics at. cores, TV bodies, etc.
T. e. interpreted on the basis of (see QUANTUM MECHANICS). Classic ch-tsa cannot be inside the potential. barrier height V, if its energy? impulse p - imaginary quantity (m - h-tsy). However, for a microparticle this conclusion is unfair: due to the uncertainty relationship, the particle is fixed in space. area inside the barrier makes its momentum uncertain. Therefore, there is a non-zero probability of detecting a microparticle inside a particle that is forbidden from the classical point of view. mechanics area. Accordingly, a definition appears. probability of passage through the potential. barrier, which corresponds to T. e. This probability is greater, the smaller the mass of the substance, the narrower the potential. barrier and the less energy is missing to reach the height of the barrier (the smaller the difference V-?). Probability of passing through a barrier - Ch. factor determining physical characteristics T. e. In the case of one-dimensional potential. such a characteristic of the barrier is the coefficient. barrier transparency, equal to ratio the flow of particles passing through it to the flow falling on the barrier. In the case of a three-dimensional barrier limiting closed area pr-va from lower. potential energy (potential well), i.e. characterized by the probability w of an individual leaving this area in units. time; the value of w is equal to the product of the frequency of oscillations inside the potential. pits on the probability of passing through the barrier. The possibility of “leakage” out of the tea that was originally in the potential. hole, leads to the fact that corresponding h-ts acquire a finite width of the order of ћw, and these themselves become quasi-stationary.
An example of the manifestation of T. e. in at. physics can serve atoms in strong electric. and ionization of an atom in a strong electromagnetic field. waves. T. e. underlies the alpha decay of radioactive nuclei. Without T. e. it would be impossible to flow thermonuclear reactions: Coulomb potential. the barrier that prevents the convergence of reactant nuclei necessary for fusion is overcome partly due to the high speed (high temperature) of such nuclei, and partly due to thermal energy. There are especially numerous examples of the manifestation of T. e. in physics TV. bodies: field emission, phenomena in the contact layer at the boundary of two PPs, Josephson effect, etc.
Physical encyclopedic Dictionary. - M.: Soviet Encyclopedia. . 1983 .
TUNNEL EFFECT
(tunneling) - systems through a movement area prohibited by classical mechanics. A typical example of such a process is the passage of a particle through potential barrier when her energy
less than the height of the barrier. Particle momentum R in this case, determined from the relation 
Where U(x)- potential particle energy ( T - mass), would be in the region inside the barrier, an imaginary quantity. IN quantum mechanics thanks to uncertainty relationship between the impulse and the coordinate, the subbarrier turns out to be possible. The wave function of a particle in this region decays exponentially, and in the quasiclassical case (see Semiclassical approximation)its amplitude at the point of exit from under the barrier is small.
One of the formulations of problems about the passage of potential. barrier corresponds to the case when a stationary flow of particles falls on the barrier and it is necessary to find the value of the transmitted flow. For such problems, a coefficient is introduced. barrier transparency (tunnel transition coefficient) D, equal to the ratio of the intensities of the transmitted and incident flows. From the time reversibility it follows that the coefficient. transparency for transitions in "direct" and reverse directions are the same. In the one-dimensional case, coefficient. transparency can be written as
integration is carried out over a classically inaccessible region, X 1,2 - turning points determined from the condition At turning points in the classical limit. mechanics, the momentum of the particle becomes zero. Coef. D 0 requires for its definition exact solution quantum-mechanical tasks.
If the condition of quasiclassicality is satisfied
along the entire length of the barrier, with the exception of the immediate neighborhoods of turning points x 1,2 .
coefficient D 0 is slightly different from one. Creatures difference D 0 from unity can be, for example, in cases where the potential curve. energy from one side of the barrier goes so steeply that the quasi-classical not applicable there, or when the energy is close to the barrier height (i.e., the exponent expression is small). For a rectangular barrier height U o and width A coefficient transparency is determined by the file
Where 
The base of the barrier corresponds zero energy. In quasiclassical case D small compared to unity.
Dr. The formulation of the problem of the passage of a particle through a barrier is as follows. Let the particle in the beginning moment in time is in a state close to the so-called. stationary state, which would happen with an impenetrable barrier (for example, with a barrier raised away from potential well to a height greater than the energy of the emitted particle). This state is called quasi-stationary. Likewise stationary states the dependence of the wave function of a particle on time is given in this case by the multiplier 
The complex quantity appears here as energy E, the imaginary part determines the probability of decay of a quasi-stationary state per unit time due to T. e.:
In quasiclassical When approaching, the probability given by f-loy (3) contains an exponential. factor of the same type as in-f-le (1). In the case of a spherically symmetric potential. barrier is the probability of decay of a quasi-stationary state from orbits. quantum number l determined by f-loy
Here r 1,2 are radial turning points, the integrand in which is equal to zero. Factor w 0 depends on the nature of the movement in the classically allowed part of the potential, for example. he is proportional. classic frequency of particle oscillations between the walls of the barrier.
T. e. allows us to understand the mechanism of a-decay of heavy nuclei. Between the -particle and the daughter nucleus there is an electrostatic force. repulsion determined by f-loy At small distances of the order of size A the nuclei are such that eff. can be considered negative: 
As a result, the probability A-decay is given by the relation
Here is the energy of the emitted a-particle.
T. e. determines the possibility of thermonuclear reactions occurring in the Sun and stars at temperatures of tens and hundreds of millions of degrees (see. Evolution of stars),
and also in terrestrial conditions in the form thermonuclear explosions or UTS.
In a symmetric potential, consisting of two identical wells separated by a weakly permeable barrier, i.e. leads to interference of states in wells, which leads to weak double splitting of discrete energy levels (so-called inversion splitting; see Molecular spectra). For an infinitely periodic set of holes in space, each level turns into a zone of energies. This is the mechanism for the formation of narrow electron energies. zones in crystals with strong connection electrons with lattice sites.
If an electric current is applied to a semiconductor crystal. field, then the zones of allowed electron energies become inclined in space. Thus, the post level electron energy crosses all zones. Under these conditions, the transition of an electron from one energy level becomes possible. zones to another due to T. e. The classically inaccessible area is the zone of forbidden energies. This phenomenon is called. Zener breakdown. Quasiclassical the approximation corresponds here to a small value of electrical intensity. fields. In this limit, the probability of a Zener breakdown is determined basically. exponential, in the cut indicator there is a large negative. a value proportional to the ratio of the width of the forbidden energy. zone to the energy gained by an electron in an applied field at a distance equal to the size of the unit cell.
A similar effect appears in tunnel diodes, in which the zones are inclined due to semiconductors R- And n-type on both sides of the border of their contact. Tunneling occurs due to the fact that in the zone where the charge carrier goes there is a finite amount of unoccupied states.
Thanks to T. e. electric possible between two metals separated by a thin dielectric. partition. These can be in both normal and superconducting state. IN the latter case may take place Josephson effect.
T. e. Such phenomena occurring in strong electric currents are due. fields, such as autoionization of atoms (see Field ionization)And auto-electronic emissions from metals. In both cases, electric the field forms a barrier of finite transparency. The stronger the electric field, the more transparent the barrier and the stronger the electron current from the metal. Based on this principle scanning tunneling microscope - a device that measures tunnel current from different points of the surface under study and providing information about the nature of its heterogeneity.
T. e. is possible not only in quantum systems consisting of a single particle. So, for example, the low-temperature movement of dislocations in crystals can be associated with tunneling of the final part, consisting of many particles. In problems of this kind, a linear dislocation can be represented as an elastic string, initially lying along the axis at in one of the local minima of the potential V(x, y). This potential does not depend on y, and its relief along the axis X is a sequence of local minima, each of which is lower than the other by an amount depending on the mechanical force applied to the crystal. voltage. The movement of a dislocation under the influence of this stress is reduced to tunneling into an adjacent minimum defined. segment of a dislocation with subsequent pulling of its remaining part there. The same kind of tunnel mechanism may be responsible for the movement charge density waves in the Peierls dielectric (see Peierls transition).
To calculate the tunneling effects of such multidimensional quantum systems, it is convenient to use semiclassical methods. representation of the wave function in the form 
Where S- classic systems. For T. e. the imaginary part is significant S, determining the attenuation of the wave function in a classically inaccessible region. To calculate it, the method of complex trajectories is used.
Quantum particle, overcoming the potential. barrier may be connected to the thermostat. In classic Mechanically, this corresponds to motion with friction. Thus, to describe tunneling it is necessary to use a theory called dissipative quantum mechanics. Considerations of this kind must be used to explain the finite lifetime of current states of Josephson contacts. In this case, tunneling occurs. quantum particle through the barrier, and the role of a thermostat is played by electrons.
Lit.: Landau L. D., Lifshits E. M., Quantum, 4th ed., M., 1989; Ziman J., Principles of Solid State Theory, trans. from English, 2nd ed., M., 1974; Baz A. I., Zeldovich Ya. B., Perelomov A. M., Scattering, reactions and decays in nonrelativistic quantum mechanics, 2nd ed., M., 1971; Tunnel phenomena in solids ah, lane from English, M., 1973; Likharev K.K., Introduction to the dynamics of Josephson junctions, M., 1985. B. I. Ivlev.
Physical encyclopedia. In 5 volumes. - M.: Soviet Encyclopedia. Chief Editor A. M. Prokhorov. 1988 .
See what the "TUNNEL EFFECT" is in other dictionaries:
Modern encyclopedia
Passage of a microparticle whose energy is less than the height of the barrier through a potential barrier; quantum effect, clearly explained by the scatter of momenta (and energies) of the particle in the barrier region (see Uncertainty principle). As a result of the tunnel... ... Big Encyclopedic Dictionary
Tunnel effect- TUNNEL EFFECT, the passage through a potential barrier of a microparticle whose energy is less than the height of the barrier; quantum effect, clearly explained by the scatter of momenta (and energies) of the particle in the barrier region (due to the uncertainty of the principle) ... Illustrated Encyclopedic Dictionary
tunnel effect- - [Ya.N.Luginsky, M.S.Fezi Zhilinskaya, Yu.S.Kabirov. English-Russian dictionary of electrical engineering and power engineering, Moscow, 1999] Topics of electrical engineering, basic concepts EN tunnel effect ... Technical Translator's Guide
TUNNEL EFFECT- (tunneling) a quantum mechanical phenomenon that consists in overcoming a potential potential (see) by a microparticle when its total energy is less than the height of the barrier. T. e. is caused by the wave properties of microparticles and affects the flow of thermonuclear... ... Big Polytechnic Encyclopedia
Quantum mechanics ... Wikipedia
Passage of a microparticle whose energy is less than the height of the barrier through a potential barrier; quantum effect, clearly explained by the spread of momenta (and energies) of the particle in the barrier region (see Uncertainty principle). As a result of the tunnel... ... encyclopedic Dictionary
1.7. The concept of the tunnel effect.
The tunnel effect is the passage of particles through a potential barrier due to wave properties particles.
Let a particle moving from left to right encounter a potential barrier of height U 0 and width l. According to classical concepts, a particle passes unhindered over a barrier if its energy E greater than the barrier height ( E> U 0 ). If the particle energy is less than the barrier height ( E< U 0 ), then the particle is reflected from the barrier and begins to move in the opposite direction; the particle cannot penetrate through the barrier.
Quantum mechanics takes into account the wave properties of particles. For a wave, the left wall of the barrier is the boundary of two media, at which the wave is divided into two waves - reflected and refracted. Therefore, even with E> U 0 it is possible (albeit with a small probability) that a particle is reflected from the barrier, and when E< U 0 there is a nonzero probability that the particle will be on the other side of the potential barrier. In this case, the particle seemed to “pass through a tunnel.”
Let's decide the problem of a particle passing through a potential barrier for the simplest case of a one-dimensional rectangular barrier, shown in Fig. 1.6. The shape of the barrier is specified by the function
. (1.7.1)
Let us write the Schrödinger equation for each of the regions: 1( x<0 ), 2(0< x< l) and 3( x> l):
;
(1.7.2)
; (1.7.3)
. (1.7.4)
Let's denote
(1.7.5)
. (1.7.6)
General solutions of equations (1), (2), (3) for each of the areas have the form:
Solution of the form 
corresponds to a wave propagating in the direction of the axis x, A 
- a wave propagating in the opposite direction. In region 1 term 
describes a wave incident on a barrier, and the term 
- wave reflected from the barrier. In region 3 (to the right of the barrier) there is only a wave propagating in the x direction, so 
.
The wave function must satisfy the continuity condition, therefore solutions (6), (7), (8) at the boundaries of the potential barrier must be “stitched”. To do this, we equate the wave functions and their derivatives at x=0 And x = l:
;
;
;
. (1.7.10)
Using (1.7.7) - (1.7.10), we obtain four equations to determine five coefficients A 1 , A 2 , A 3 ,IN 1 And IN 2 :
A 1 +B 1 =A 2 +B 2 ;
A 2 exp( l) + B 2 exp(- l)= A 3 exp(ikl) ;
ik(A 1 - IN 1 ) = (A 2 -IN 2 ) ; (1.7.11)
(A 2 exp( l)-IN 2 exp(- l) = ikA 3 exp(ikl) .
To obtain the fifth relation, we introduce the concepts of reflection coefficients and barrier transparency.
Reflection coefficient let's call the relation
, (1.7.12)
which defines probability reflection of a particle from a barrier.
Transparency factor
(1.7.13)
gives the probability that the particle will pass through the barrier. Since the particle will either be reflected or pass through the barrier, the sum of these probabilities is equal to one. Then
R+ D =1; (1.7.14)
. (1.7.15)
That's what it is fifth relationship that closes the system (1.7.11), from which all five coefficients
Of greatest interest is transparency coefficientD. After transformations we get
,
(7.1.16)
Where D 0 – value close to unity.
From (1.7.16) it is clear that the transparency of the barrier strongly depends on its width l, on how high the barrier is U 0 exceeds the particle energy E, and also on the mass of the particle m.
WITH 
from the classical point of view, the passage of a particle through a potential barrier at E<
U 0
contradicts the law of conservation of energy. The fact is that if a classical particle were at some point in the barrier region (region 2 in Fig. 1.7), then its total energy would be less than the potential energy (and the kinetic energy would be negative!?). WITH quantum dot there is no such contradiction. If a particle moves towards a barrier, then before colliding with it it has a very specific energy. Let the interaction with the barrier last for a while
t, then, according to the uncertainty relation, the energy of the particle will no longer be definite; energy uncertainty 
. When this uncertainty turns out to be on the order of the height of the barrier, it ceases to be an insurmountable obstacle for the particle, and the particle will pass through it.
The transparency of the barrier decreases sharply with its width (see Table 1.1.). Therefore, particles can pass through only very narrow potential barriers due to the tunneling mechanism.
Table 1.1
Values of the transparency coefficient for an electron at ( U 0 – E ) = 5 eV = const
|
l, nm | |||||
We considered a rectangular shaped barrier. In the case of a potential barrier of arbitrary shape, for example, as shown in Fig. 1.7, the transparency coefficient has the form
. (1.7.17)
The tunnel effect manifests itself in a number of physical phenomena and has important practical applications. Let's give some examples.
1. Field electron (cold) emission of electrons.
IN 
In 1922, the phenomenon of cold electron emission from metals under the influence of a strong external electric field was discovered. Potential Energy Graph U electron from coordinate x shown in Fig. At x
<
0 is the region of the metal in which electrons can move almost freely. Here potential energy can be considered constant. A potential wall appears at the metal boundary, preventing the electron from leaving the metal; it can do this only by acquiring additional energy, equal to work exit A.
Outside the metal (at x >
0) the energy of free electrons does not change, so when x> 0 the graph U(x)
goes horizontally. Let us now create a strong electric field near the metal. To do this, take a metal sample in the shape of a sharp needle and connect it to the negative pole of the source. Rice. 1.9 Operating principle of a tunnel microscope
ka voltage, (it will be the cathode); We will place another electrode (anode) nearby, to which we will connect the positive pole of the source. If the potential difference between the anode and the cathode is large enough, it is possible to create an electric field with a strength of about 10 8 V/m near the cathode. The potential barrier at the metal-vacuum interface becomes narrow, electrons leak through it and leave the metal.
Field emission was used to create vacuum tubes with cold cathodes (they are now practically out of use); it has now found application in tunnel microscopes, invented in 1985 by J. Binning, G. Rohrer and E. Ruska.
In a tunnel microscope, a probe - a thin needle - moves along the surface under study. The needle scans the surface under study, being so close to it that electrons from the electron shells (electron clouds) of surface atoms, due to wave properties, can reach the needle. To do this, we apply a “plus” from the source to the needle, and a “minus” to the sample under study. The tunnel current is proportional to the transparency coefficient of the potential barrier between the needle and the surface, which, according to formula (1.7.16), depends on the barrier width l. When scanning the surface of a sample with a needle, the tunneling current varies depending on the distance l, repeating the surface profile. Precision movements of the needle over short distances are carried out using the piezoelectric effect; for this, the needle is fixed on a quartz plate, which expands or contracts when an electrical voltage is applied to it. Modern technologies make it possible to produce a needle so thin that there is only one atom at its end.
AND 
the image is formed on the computer display screen. Permission tunnel microscope so high that it allows you to “see” the arrangement of individual atoms. Figure 1.10 shows an example image of the atomic surface of silicon.
2. Alpha radioactivity ( – decay). In this phenomenon, a spontaneous transformation of radioactive nuclei occurs, as a result of which one nucleus (it is called the mother nucleus) emits an particle and turns into a new (daughter) nucleus with a charge less than 2 units. Let us recall that the particle (the nucleus of a helium atom) consists of two protons and two neutrons.
E 
If we assume that the α-particle exists as a single formation inside the nucleus, then the graph of the dependence of its potential energy on the coordinate in the field of the radioactive nucleus has the form shown in Fig. 1.11. It is determined by the energy of the strong (nuclear) interaction, caused by the attraction of nucleons to each other, and the energy of the Coulomb interaction (electrostatic repulsion of protons).
As a result, is a particle in the nucleus with energy E is located behind the potential barrier. Due to its wave properties, there is some probability that the particle will end up outside the nucleus.
3. Tunnel effect inp- n- transition used in two classes of semiconductor devices: tunnel And reversed diodes. A feature of tunnel diodes is the presence of a falling section on the direct branch of the current-voltage characteristic - a section with a negative differential resistance. The most interesting thing about reverse diodes is that when connected in reverse, the resistance is less than when connected in reverse. For more information on tunnel and reverse diodes, see section 5.6.
Tunnel effect
Tunneling effect
Tunnel effect (tunneling) – the passage of a particle (or system) through a region of space in which stay is prohibited classical mechanics. Most famous example such a process is the passage of a particle through a potential barrier when its energy E is less than the barrier height U 0 . In classical physics, a particle cannot appear in the region of such a barrier, much less pass through it, since this violates the law of conservation of energy. However, in quantum physics the situation is fundamentally different. A quantum particle does not move along any specific path. Therefore, we can only talk about the probability of finding a particle in a certain region of space ΔрΔх > ћ. In this case, neither potential nor kinetic energies have definite values in accordance with the uncertainty principle. A deviation from the classical energy E by the amount ΔE is allowed during time intervals t given by the uncertainty relation ΔEΔt > ћ (ћ = h/2π, where h – Planck's constant).
The possibility of a particle passing through a potential barrier is due to the requirement of continuous wave function on the walls of the potential barrier. The probability of detecting a particle on the right and left is related to each other by a relationship that depends on the difference E - U(x) in the region of the potential barrier and on the barrier width x 1 - x 2 at a given energy.
As the height and width of the barrier increases, the probability of a tunnel effect decreases exponentially. The probability of a tunnel effect also decreases rapidly with increasing particle mass.
Penetration through the barrier is probabilistic. Particle with E< U 0 , натолкнувшись на барьер, может либо пройти сквозь него,
либо отразиться. Суммарная вероятность этих двух возможностей равна 1. Если
на барьер падает поток частиц с Е < U 0 , то часть этого
потока будет просачиваться сквозь барьер, а часть – отражаться. Туннельное
прохождение частицы через потенциальный барьер лежит в основе многих явлений
ядерной и atomic physics: alpha decay, cold emission of electrons from metals, phenomena in the contact layer of two semiconductors, etc.
> Quantum tunneling
Explore quantum tunnel effect . Find out under what conditions the tunnel vision effect occurs, Schrödinger's formula, probability theory, atomic orbitals.
If an object does not have enough energy to break through the barrier, then it is able to tunnel through an imaginary space on the other side.
Learning Objective
- Identify factors influencing the probability of tunneling.
Main points
- Quantum tunneling is used for any objects in front of the barrier. But for macroscopic purposes the probability of occurrence is small.
- The tunnel effect arises from Schrödinger's imaginary component formula. Since it is present in the wave function of any object, it can exist in imaginary space.
- Tunneling decreases as the body mass increases and the gap between the energies of the object and the barrier increases.
Term
- Tunneling is the quantum mechanical passage of a particle through an energy barrier.
How does the tunnel effect occur? Imagine throwing a ball, but it disappears instantly without ever touching the wall, and appears on the other side. The wall here will remain intact. Surprisingly, there is a finite probability that this event will come to fruition. The phenomenon is called the quantum tunneling effect.
At the macroscopic level, the possibility of tunneling remains negligible, but is consistently observed at the nanoscale. Let's look at an atom with a p orbital. Between the two lobes there is a nodal plane. There is a possibility that an electron can be found at any point. However, electrons move from one lobe to another by quantum tunneling. They simply cannot be in the hub area, and they travel through an imaginary space.
The red and blue lobes show volumes where there is a 90% probability of finding an electron at any time interval if the orbital zone is occupied
Temporal space does not appear to be real, but it actively participates in Schrödinger’s formula:
All matter has a wave component and can exist in imaginary space. A combination of the object's mass, energy, and energy height will help understand the difference in tunneling probability.
As the object approaches the barrier, the wave function changes from sine wave to exponentially contracting. Schrödinger formula:
The probability of tunneling becomes less as the mass of the object increases and the gap between energies increases. The wave function never approaches 0, which is why tunneling is so common at nanoscales.
(solving the problems of the PHYSICS block, as well as other blocks, will allow you to select THREE people for the full-time round who scored in solving the problems of THIS block greatest number points. Additionally, based on the results of the head-to-head round, these candidates will compete for a special nomination “ Physics of nanosystems" Another 5 people with the highest scores will also be selected for the full-time round. absolute number of points, so after solving problems in your specialty there is full meaning to solve problems from other blocks. )
One of the main differences between nanostructures and macroscopic bodies is the dependence of their chemical and physical properties from size. A clear example This is achieved by the tunnel effect, which consists in the penetration of light particles (electrons, protons) into areas that are energetically inaccessible to them. This effect plays important role in processes such as charge transfer in photosynthetic devices of living organisms (it is worth noting that biological reaction centers are among the most effective nanostructures).
The tunnel effect can be explained by the wave nature of light particles and the uncertainty principle. Due to the fact that small particles do not have a specific position in space, there is no concept of trajectory for them. Consequently, to move from one point to another, a particle does not have to pass along the line connecting them, and thus can “bypass” energy-forbidden regions. Due to the absence of an exact coordinate for an electron, its state is described using a wave function that characterizes the probability distribution along the coordinate. The picture shows typical look wave function when tunneling under an energy barrier.
Probability p penetration of an electron through a potential barrier depends on the height U and the width of the latter l ( Formula 1, left), Where m– electron mass, E– electron energy, h – Planck’s constant with a bar.
1. Determine the probability that an electron tunnels to a distance of 0.1 nm if the energy differenceU –E = 1 eV ( 2 points). Calculate the energy difference (in eV and kJ/mol) at which an electron can tunnel a distance of 1 nm with a probability of 1% ( 2 points).
One of the most noticeable consequences of the tunnel effect is the unusual dependence of the rate constant chemical reaction on temperature. As the temperature decreases, the rate constant tends not to 0 (as can be expected from the Arrhenius equation), but to constant value, which is determined by the probability of nuclear tunneling p( f formula 2, left), where A– pre-exponential factor, E A – activation energy. This can be explained by the fact that when high temperatures Only those particles whose energy is higher than the barrier energy enter into the reaction, and when low temperatures the reaction occurs solely due to the tunnel effect.
2. From the experimental data below, determine the activation energy and tunneling probability ( 3 points).
|
k(T), c – 1 |
|
In modern quantum electronic devices The effect of resonant tunneling is used. This effect occurs if an electron encounters two barriers separated by a potential well. If the electron energy coincides with one of the energy levels in the well (this is the resonance condition), then overall probability Tunneling is determined by passing through two thin barriers, but if not, then a wide barrier appears in the path of the electron, which includes a potential well, and the overall probability of tunneling tends to 0.
3. Compare the probabilities of resonant and non-resonant electron tunneling for the following parameters: the width of each barrier is 0.5 nm, the width of the well between the barriers is 2 nm, the height of all potential barriers relative to the electron energy is 0.5 eV ( 3 points). Which devices use the tunneling principle ( 3 points)?